Pythagoras' Theorem

What if you could find a missing side of a triangle using just two numbers? Meet our right triangle with legs three and four — we want the hypotenuse. The vertical leg is called a, shown here in blue.

The horizontal leg is b, shown in green. And the longest side, opposite the right angle, is the hypotenuse c in orange. Now let's build a square on each side and count the unit squares inside.

The square on side b fills up with sixteen unit squares. The square on the hypotenuse c has exactly twenty-five unit squares. Nine plus sixteen equals twenty-five — the two smaller squares fill the big one perfectly!

• 9 + 16 = 25 ✓

That pattern is always true for any right triangle — here is the formula.

a squared is the area of the blue square, b squared is the green square. And c squared is the area of the orange square on the hypotenuse. Let's solve our example step by step — substitute a equals three and b equals four.

Three squared is nine, four squared is sixteen.

Nine plus sixteen gives us twenty-five equals c squared.

Taking the square root of both sides, c equals five exactly.

Now let's see WHY this is true with a beautiful visual proof. The big square has four identical triangles arranged around a tilted square of area c squared. Watch what happens when we slide the triangles into the other arrangement.

The same four triangles now surround two squares — one of area a squared and one of area b squared. So the gaps must be equal, proving a squared plus b squared always equals c squared. A very common mistake — students add the sides instead of squaring them first.

• Same triangles → same gaps!

Always square first, then add, then take the square root at the very end. Here is a real-life use — a ladder leaning against a wall forms a right triangle. If the wall is six metres and the ground distance is eight metres, the ladder is ten metres.

• Square → Add → Root

Now your turn — a right triangle has legs five and twelve, what is the hypotenuse? Remember — in any right triangle, the two smaller squares always add up to the biggest square.